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Theorem r0cld 23762
Description: The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
r0cld ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld
StepHypRef Expression
1 kqval.2 . . . . . 6 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqffn 23749 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1132 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 Fn 𝑋)
4 fncnvima2 7081 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}})
53, 4syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}})
6 fvex 6920 . . . . . 6 (𝐹𝑧) ∈ V
76elsn 4646 . . . . 5 ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ (𝐹𝑧) = (𝐹𝐴))
8 simpl1 1190 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 simpr 484 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝑧𝑋)
10 simpl3 1192 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐴𝑋)
111kqfeq 23748 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
12 eleq2w 2823 . . . . . . . . 9 (𝑦 = 𝑜 → (𝑧𝑦𝑧𝑜))
13 eleq2w 2823 . . . . . . . . 9 (𝑦 = 𝑜 → (𝐴𝑦𝐴𝑜))
1412, 13bibi12d 345 . . . . . . . 8 (𝑦 = 𝑜 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑜𝐴𝑜)))
1514cbvralvw 3235 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜))
1611, 15bitrdi 287 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
178, 9, 10, 16syl3anc 1370 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
187, 17bitrid 283 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
1918rabbidva 3440 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}} = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
205, 19eqtrd 2775 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
211kqid 23752 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22213ad2ant1 1132 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23 simp2 1136 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ Fre)
24 simp3 1137 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐴𝑋)
25 fnfvelrn 7100 . . . . . 6 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
263, 24, 25syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
271kqtopon 23751 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28273ad2ant1 1132 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29 toponuni 22936 . . . . . 6 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = (KQ‘𝐽))
3028, 29syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → ran 𝐹 = (KQ‘𝐽))
3126, 30eleqtrd 2841 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ (KQ‘𝐽))
32 eqid 2735 . . . . 5 (KQ‘𝐽) = (KQ‘𝐽)
3332t1sncld 23350 . . . 4 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝐴) ∈ (KQ‘𝐽)) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
3423, 31, 33syl2anc 584 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35 cnclima 23292 . . 3 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3622, 34, 35syl2anc 584 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3720, 36eqeltrrd 2840 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  ran crn 5690  cima 5692   Fn wfn 6558  cfv 6563  (class class class)co 7431  TopOnctopon 22932  Clsdccld 23040   Cn ccn 23248  Frect1 23331  KQckq 23717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-qtop 17554  df-top 22916  df-topon 22933  df-cld 23043  df-cn 23251  df-t1 23338  df-kq 23718
This theorem is referenced by:  nrmr0reg  23773
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